one can use the summation formula [1]} to obtain that \(\,\mathrm {e}^{- \frac{s}{{E}_B}H_B}\, = \pi (g_s)\) , with
\(g_s(x) := \frac{\,\mathrm {e}^{-\frac{|x|^2}{4\ell _B^2}\coth \left(\frac{s}{2}\right)}\,}{2\sinh \left(\frac{s}{2}\right)} \in {S}\big (\mathbb {R}^2\big ) .\) 
